Problem 3.14 (Schroeder’s Intro to Thermal Physics)

Problem 3.14

Experimental measurements of the heat capacity of aluminum at low temperatures (below about 50 K) can be fit to the formula

$C_V=aT+bT^3,$

where $C_V$ is the heat capacity of one mole of aluminum, and the constants $a$ and $b$ are approximately $a = 0.00135 \text{J}/\text{K}^2$ and $b = 2.48 \times 10^{-5} \text{J}/ \text{K}^4.$ From this data, find a formula for the entropy of a mole of aluminum as a function of temperature. Evaluate your formula at T = 1 K and at T = 10 K, expressing your answers both in conventional units (J/K) and as unitless numbers (dividing by Boltzmann’s constant). [Comment: In Chapter 7 I’ll explain why the heat capacity of a metal has this form. The linear term comes from energy stored in the conduction electrons, while the cubic term comes from lattice vibrations of the crystal.]